I. The Phase-Current Relation at Zero Vo

I. The Phase-Current Relation at Zero Voltage in Proximity Effect Bridges. II. The Interaction of Proximity Effect Bridges with Superconducting Microstrip Resonators - CaltechTHESIS
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I. The Phase-Current Relation at Zero Voltage in Proximity Effect Bridges. II. The Interaction of Proximity Effect Bridges with Superconducting Microstrip Resonators
Citation
Ganz, Tomas
(1976)
I. The Phase-Current Relation at Zero Voltage in Proximity Effect Bridges. II. The Interaction of Proximity Effect Bridges with Superconducting Microstrip Resonators.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/N97W-BW02.
Abstract
Experimental investigations on the proximity effect bridge (a Josephson device) at zero voltage and at finite voltages in the µV range are reported.
The phase-super current relation at zero voltage was measured using an asymmetric superconducting quantum
interferormeter circuit. The data are in agreement with the Josephson supercurrent-phase relation I_S=I_C sinδ with deviation less than 5% of the critical current I_c. The supercurrent density in the measured bridges reached as high as 50-100 µA/µm^2.
Using microcircuitry techniques, proximity effect bridges
were strongly coupled to superconducting microstrip resonators. Selfinduced steps in the I-V characteristics of bridges coupled to resonators were observed in the GHz region at voltages (frequencies) corresponding to the expected modes of the resonators. Two types of steps were
seen depending on whether the resonator impedance on resonance was much higher or much smaller than the bridge resistance. A simple two fluid model of the bridge-resonator circuit was developed and the size and shape of self-induced steps were calculated for a generalized
Josephson oscillator relation I_S = I_c(l-q + q sin∫ 2e/hV dt) where q = 1 corresponds to the original Josephson relation and q = 1/2 represents the phase slip regime. At low critical currents (I_c < 10 µA) and low voltages (V < 3µV) the size and shape of experimentally observed
self-induced steps agree with the q = 1 model. At higher voltages and/or critical currents the step size increasingly deviates from the q = 1 model towards q = 1/2. These observations are interpreted to indicate a progressive reduction of the amplitude of the oscillating
Josephson supercurrent in proximity effect bridges from I_c towards I_c /2 as the critical current and/or voltage are increased.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics)
Degree Grantor:
California Institute of Technology
Division:
Physics, Mathematics and Astronomy
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Mercereau, James E.
Thesis Committee:
Unknown, Unknown
Defense Date:
5 March 1976
Record Number:
CaltechTHESIS:12062011-081306612
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DOI:
10.7907/N97W-BW02
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THE PHASE-CURRENT RELATION AT ZERO VOLTAGE
IN PROXIMITY EFFECT BRIDGES

II.

THE IN TERACTION OF PROXIMITY EFFECT BRIDGES
WITH SUPERCONDUCTING MICROS TRIP RESONATORS

Thesis by
Tomas Ganz

In partial fulfillm e nt of the requirements
for the degr e e of
Docto r of Philosophy

California Institute of Technolo gy
Pasadena,

California

1975

(Submi.tted March 5;

1976)

-11-

To Patti and our parents.

-111-

ACKNOWLEDGEMENTS
This work could not have been undertaken without the intellectual,
te c hnical and finan c i a l assistance from many sources .
bution is gratefully acknowledged.

Their contri-

By necessity, only a partial listin g

is given.
Dr. Ja"m es Me r ce reau who provided the leadership and
direction which ma de the Low Temperature Physics group at Caltech
an exciting place to learn physics,
Dr. Harris Notarys who tried to teach me his own perfectionist
brand of exper imental physi c s and who was involved in the development
of all the technology used in my experiments (I cannot thank him
enough),
Dr. G . John Dick who exposed "m e to resonators and COIl1.puter
programming and whose insights into experimental problems sav ed me
from many a blind alle y,
Bob MacNamara who cooperated closely on so"m e of the experim e nts with bridge-resonator circuits and who will carryon when I
l eave,
The fellow graduate students in weak superconductivity Dave
Palmer, Steve Decker, Ming Lun Yu and Run Han 'Wong whose hard
work and inventiveness broke the ground for my own efforts and whose
friendship and good humor was much appreciated,
Sandy Santantonio and Ed Boud who help.ed design and make
111.uch of the equipment neede d in our work and w ho se meticulous
craftsmanship set a high standard for all of us,

-lV-

Gail Kusudo ,vho made me look forward to lunch even when the
experiments were going well and assisted with many organizational
'matters, order s,manus cripts and travel,
Karen Cheetham whose intelligent and resourceful typing from
IUY me s sy manus cript is

an adluirable accomplishment,

IBM Corporation who awarded 'me a predoctoral fellowship

(1973-4),
The California Institute of Technology which generously contributedmuch financial and logistical support and a highly creative
environment which I will sorely mis s.

-v-

ABSTRACT
Experimental investigations on the proximity effect bridge (a
Josephson device) at zero voltage and at finite voltages in the fLY range
are reported.
The phase-super current relation at zero voltage was measured
using an asym.rnetric superconducting quantum interferorneter circuit.
The data are in agreen1.ent with the Josephson supercurrent-phase
relation
IS = I c sin 6 with deviation less than 5% of the critical current
Ic'

The supercurrent density in the rneasured bridges reached as

hi gh as 50 -1 00 fLA / fLrn .
Using rnicrocircuitry techniques, proxirnity effect bridges
were strongly coupled to super conducting Inicrostrip resonators.

Self-

induced steps in the I-V characteristics of bridges coupled to resonators
were observed in the GHz region at voltages (frequencies) corresponding
to the expected rnodes of the resonators.

Two types of steps were

seen depending on whether the resonator irnpedance on resonance was
rnuch higher or rnuch smaller than the bridge resistance.

A sirnple

two fluid rnodel of the bridge-resonator circuit was developed and the
size and shape of self-induced steps were calculated for a generalized
Josephson oscillator relation IS = Ic(l-q + q sinJ 2ii V dt) where q = 1
corresponds to the original Josephson relation and q = 1/2 represents
the phase slip regirne.

At low critical currents (I

< 10 fLA) and low

voltages (V < 3fLV) the size and sha·pe of experimentally observed
self-induced steps agree with the q = 1 rnodel.

At higher voltages

and/or critical currents the step size incr easingly deviates frorn the

-viq = 1 nlOdel towards q = 1/2.

These observations are interpreted to

indicate a progressive reduction of the amplitude of the oscillating
Josephson supercurrent in proximity effect bridges froTil I
I /2 as the critical current and/or voltage are increased.

towards

-viiTABLE OF CONTENTS
Page
Acknowledgements

Abstract

Introduction

II.

Macroscopic QuantUln Effects - Theory

1.1

The Macroscopic Wavefunction

1.2

The Boundary of a Superconductor

1 .3

Junctions in the Weak Coupling LiInit

1.4

The Phase Slip Model

1.5

The Current-Voltage Charact eristic of
Jo sephson D evices

1 .6

Quantum Interference Effects in
Magnetic Fields

References

The Phase-Current Relation at Zero Voltage in
Proximity Effect Bridges

2.1

Historical Background

2.2

The Small Asymmetric Interferometer

2 .3

Sample Preparation

2.4

The Measurement of the Int erferometer
Criti cal Current as a Function of Applied
Magnetic Field

2 .5

Data Analysis,

2.6

Conclusion

References

Results and Discussion

-viiiTABLE OF CONTENTS (Cont'd)
Page
III.

The Interaction of Proxim.ity Effect Bridges
with Sup erconducting 11icro strip Resonator s

3.1

Introducti on

3.2

Self-induced Steps -- General Considerations

3. 3

Experimental Technique

3.4

Obs ervations

3.5

Self-induced Steps for a Simple Harmonic
Phase -Supercurrent Relation
Theory

101

3.6

Results and Discussion

110

3. 7

Conclusion

114

References

124

-1INTRODUCTION
The study of Josephson phenomena in weakly coupled superconducto r s has adv anced greatly since its beginning in 1962.

Chapter

1 of this thesis catalogs the ideas which in the author's opinion forrn
the foundation of current effort in this area.

Chapter 2 and Chapter 3

report on research conducted by the author in an attempt to describe
in more detail the dynamics of the proximity effect bridge, a
Josephson device showing major protnise in applications ranging
from magnetometry to far-infrared radiation detectors.
Chapter 2 deals with the zero voltage regime in ,vhich
equilibriulu therrrlOdynamics applies.

The superconducting quantum

interferometer was used as a tool to establish that the proximity effect
bridge obeys the same equations at zero voltage as were proposed
by Josephson for the tunneling junction.
Chapter 3 covers situations in which nonequilibrium processes
are expected to cause major deviations froul the Josephson tunneling
equations.

The interaction of proxiluity effect bridges with super-

conductingmicrostrip resonators was explored both as an end in
itself and as a means to estimate the amplitude of the super current
oscillation in these bridges.

It was found that as the voltage increases

the amplitude undergoes a transition from that expected from classical
Josephson equations towards the smaller relative atnplitude expected
from the nonequilibrium phase-slip theories. ··SiIT1ilar but less marked
trend towards a relative reduction of the oscillation atnplitude was
seen with increasing critical current in these devices.

-2-

1.
1,1

MACROSCOPIC QUANTUM EFFECTS - THEORY

The Macroscopic Wavefunction
The m.acroscopic wavefunction concept was introduced by

London (Ref.

1) who proposed that superconductivity is a phase in

which electrons are condensed into a single state described by a
single wavefunction

ljJ

-+
= ljJ(r,
t),

This description was taken

further by Ginzburg and Landau (GL) (Ref. 2) with particular
attention to the tem.perature regim.e in the vicinity of the superconducting transition,
(BCS) (Ref.

In 1957,

Bardeen,

Cooper and Schrieffer

3) developed a m.icroscopic theory of superconductivity

based on the phonon mediated electron-electron interaction.

Since

both the London and GL theories are derivable from. the BCS
theory,

the latter will be used as a starting point in this

di s eus s ion.

In the BCS picture pairwise attraction between electrons
near the Fermi level occurs via the distortion of the lattice
induced by each m.em.ber of a pair.

Below a certain transition

temperature this attraction re sults in the form.ation of bound
electron pairs with antiparallel spins.

These pairs have m.any

characteristics of bosons and are condensed into a single state as
predicted by London.

According to BCS a certainminim.um.

energy is required to break up an existing pair.
designated E

tem.perature T.

The energy is

= 211(T) and is referred to as the energy gap at
The excitations resulting from. pair breaking are

called quasiparticles, and it can be shown (Ref.

4) that they

behave like electrons in a norm.al (non-superconducting) m.etal.

-3-

At finite temperatur e s,

pairs are broken by thermal agitation which

leads to a dynalnic equilibrium between quasiparticles and pairs.
The two fluids,

the pair fluid and the quasiparticle fluid can be

approximately considered as mutually noninteracting.
The condensate of electron pairs can be represented by a
London wavefunction
-+
ljJ (r, t)

i cp(i, t)
= _I....
Vp(r, t) e

(1. 1)

-+
-+
where p(r, t) is the electron pair density and cp(r, t) is the phase

of the wavefunction.

In general the pair density depends on

telnperature and pair ve l ocity and should be calculated using the
full BCS or GL theory.

Nevertheless,

at a given temperature

and velocity well below critical velocity the conde nsate obeys
simple quantu"m mechanics (R ef.

5).

Applying the quantum

"mechanical expression for electrical current density to the form
(l. l) leads to
-+

2e-li -+
2e...,
= - -(Vcp + - 1\) P
m-li

(1.2.)

-+
where A is the magnetic vector potential and -2e is the charge

of an electron pair.

For bulk simply connected superconductors,

one can work in the London gauge (Ref.
superconductor.

6) where ;5.

A = 0 i l'l the

-+
-+ -:}
Assuming p(r, t) = const. and V • J = 0 (steady

state) it can be seen that in the superconductor
V cp

(1. 3)

= 0 everywhere since on the surface of the
-+
Therefore equation (1.2) yields
superconductor (vcp )n = O.

This implies ;5cp

(1. 4 )

-4Equation (1. 4) is the London equation.

The magnetic vector

pot ential is related to its source curr e nt by

·l1. = - -\1

(1. 5)

Combining (1. 4) and (1. 5)
'V

A = 4 e 2p

(1. 6)

where A IS the London penetration depth (typ.

3 •
10 - lOA).

The

....

physical solution of equation (1. 6) is a potential A which decays
exponentially from the surface of the superconductor inward with
decay length A.

Thus the magnetic field many penetration depths

A inside the superconductor is zero (Ref.

7).

From equation

(1.4) it follows that the supercurrent density

"1 is distributed

similarly to the magnetic vector potential.
Next a piece of superconductor which is not simply
connected (e. g.,

a ring) shall be considered.

Rewriting equation

(1.2) one obtains

.'Vlfl
. = - 2em"1
_ 2eh 1.
hp

(1. 7)

Integrating once around the ring
(1. 8)

But the wavefunction (1. 1) must have only one value at a given
point which implie s that
6lfl

= 2nn

and
(1. 9)

-5-

,c""

,t2 44
J. dt

~ A· dt + fLO;r)'

where iJi

(1. 10)

is called the flux quantum.

The first left hand side

term of equation (1. 10) is just the magnetic flux through the path
of integration.

If the path is taken deep inside the superconducting

luaterial of the ring,

== 0 and the magnetic flux through the

ring is
==

niJi

(1.11)

which is referred to as "quantization of flux".
has been observed experimentally (Ref.

1.2

This phenomenon

8).

The Boundary of a Superconductor
At the boundary between a superconductor and vacuum two

processes take place.

Some electrons which are bound in pairs

in the superconductor penetrate the boundary (with roughly the
velocity v

the Fermi velocity) and "depair" (lose their pair

binding ener gy 2t::,).

Also,

the penetrating electrons find them-

selves in the negative energy region outside the superconductor
formed by the work function of the metal Wand the image
potential,

and are reflected back into the superconductor.

The

characteristic lengths for the two processes are the depairing
length

and the tunneling length d.

using the uncertainty principle with t::,

==2xlO

The se can be estimated

1 meV,

W == 5 eV,

m/sec

. 2t::,h

2000A
(1.12)

12m W

-6Therefore at a superconductor-vacuum interface the distance for
pair penetration is dominated by the tunneling length d,

Similar 1y,

at a boundary of a supe rconductor and an insulating dielectric,

the

e l ectrons leaving a superconductor encounter an energy barrier in
the form of the band gap (typically a few eV ),

The resulting

tunneling length d is of the same order as at the superconductor"vacuum interface,
At a superconductor-normal metal interface (S-N) the
situation is different.

The Fermi levels in the two metals have

equalized and single electrons are energetically free to move
between the two metals.

The decay length of the pair wavefunction

will therefore be determined by the depairing distance for Cooper
pairs,

It can be shown that this distance is (in the long mean

free path limit)

~SN
where T

(1.13)

= v F 3k B (T-T cZ )

is the super conducting transition temperature of the

nor-mal metal (i. e.,

T > T

The pair wavefunction therefore

has a finite amplitude in the normal metal due to pair diffusion
and similarly normal electrons diffuse in the opposite direction
into the superconductor,

Such normal electron diffusion diminishes

the equilibriu"m pair density in the vicinity of the boundary,
thereby decreasing the gap energy Zt,(T) as if the temperature of
th e superconductor at the boundary were raised.

This is referr e d

to as the "proximity effect".

If the S-N interface car ries a current through the interface

-7-

in either direction the equilibrium between the condensed pairs
and quasiparticles is upset,

i. e.,

their equilibrium concentration.

the injecte d particles exceed
The disequilibrium is described

in terms of electrochemical potentials for pairs f.Lp (per electron)
and quasiparticles f.LQ with definitions

2f.Lp

(1.14)

(1.15)

where cp is the phase of the macroscopic wave function and
GQ(T, PQ' ip) is th e Gibbs free energy of quasiparticle s as a
function of temperature,
potential.

quasiparticle density and electrostatic

The position dependence of electrochemical potentials

f.Lp and f.LQ near an S-N interfac e has been studied by Yu and
Mercereau (Ref.

9) who showed that while f.LQ has a gradient

across the boundary,

f.Lp stays constant .

This is reasonable

since for quasiparticles
-,}N
-J
where

'Vf.Le Q

(1.16)

is the normal conductivity of the metal,

pairs (see equation 1. 2)

whereas for

(aA/at = 0 assumed).

= 4e 2 P. ~f.Lp

(1 .17)

If the pair electro chemical potential developed a gradient,

the

supercurrent would accelerate and a steady state could not be
achieve d .

Since the total current density

across the S-N boundary,

J = JS + I N

the decay of the supercurrent

is constant

JS in

-8spa ce must correspond to an appropriate incr ease in the normal
current I

so that
(1.18)

The other assumption used in S-N interfa ce studies is the
relaxation approximation according to which the deviation from
equilibrium flQ-flp is proportional to the rate of "minority carrier"
accumulation
(1. 19)

Combining this expression with equation (1. 16) and the constancy
of flp one obtains

flO - flp

(1. 20)

A OP

where AQp is the relaxation length describing the spatial extent
of the nonequilibr ium region for mutual pair -quasiparticle
conve r s ion (Ref.

10).

The constant AOp presumably depends both

on the material and the nature of the dominant relaxation process
for pair -quasiparticle conver sion.

Since deep inside the super-

conductor both pair and quasipa rticle densities are nonzero (at
finit e temperatures) flO - fJ.p = 0 must hold there.

Deep inside

the normal metal th e tail of the pair wavefunction is broken by
thermal agitation and flp is not defined.
1. 3

Junctions in the Weak Coupling Limit
'When two piece s of superconductor are brought into

proxim.ity such that the behavior of the pair wavefunction in one
superconductor perturbs the pair wavefunction in the other

-9superconductor,

a device with many interestin g properties is

produced--a Josephson de vice .
Josephson (Ref,

The situation first considered by

11) involved two pieces of superconductor separated

by a small (= a few tunneling length s d) insulating oxide space
(typi cally 30 A thick); but as expected from the discussion i l l
section 102,

this distance may be made longer (~400 A) if the

space separating the superconductors is filled with a semiconductor
(Ref,

12)0

A S chro edinger equation for the two c oupled super-

conductors (Ref.

1 3 ) may be written as

i 11

at (tJ1 ) = (fl.l

tJ1z

wher e tJ1 ,

and tJ1

(l.Zl)

are the macroscopic wavefunctions ln supercon-

ductors 1 and 2 and K is the coupling between them,

while fl.l and

fl.Z are the electrochemical potentials (per -electron) for pairs in the
t wo superconductors.
and th e coupling K

Writing the wavefunctions in the form (l.l)

= ke ){ one obtains that

cP - at (CP2 - CPI)

2(fl.z -fl.l)
(1. ZZ)

and
PI
where PI

= -P2 = -k~PI Pz sin (cp + It)

(1. Z3)

is the initial rat e of pair density loss that would occur

if an external battery did not supply more electrons.

Let

S be

the characteristic length ove r which the pair wavefunction responds
to p er turb a tion--the coher e n c e length (Re f.
- ductor.

14) of the supercon-

Then the supercurrent flowing through a current biased

juncti on is

-10-

(1. 24)

It is customary to define

- kg~

and 6 = CP+K

so that

the Josephson equations can be written in simple form

and

2(11,1 -~2)

2 eV

= - fl- ·-

(1.25 )

where the chemical potential . fJ.

flK/2

It should be observe d that for zero v oltage V

can flow,

i.e.;

= 0 curr.ent still

the Jos ephson device superconducts.

maximum supe rcur rent density that

The

can be thus conducted is J 0'

called the critical current density of the junction.

In the absence

of a directional influence in the junction (such as magnetic field
or voltage) we may set It

= O.

In the presence of magnetic field

gauge invariance require s that
2e

where the integral is taken between the reference points for the
phase difference

If ?y some means a vo ltage V =
across the junction,

.f12 -f.l.r

is maintained

the Josephson equations indicate that the

supercurrent will oscillate at the frequency

The factor

= 2e

has the magnitude 484 MHz/fJ.V.

(1. 26)

In addition

the voltage acro s s the junction will also produce a flow of
quasiparticles across the barrier,
(1. 27)

-11~

where

is the resistan ce per square of the junction.

The total

current density through a Josephson device is then

= J o sin 6 +VIR

(1.2 8)

where

= 2eh S Vdt

In the original paper by Josephson (Ref.

(1. 29)
11) an additional so-called

pair-quasiparticle interference tenn was included.

Its size is

currently under active scrutiny in many laboratories (Ref.
1.4

IS) .

The Phase Slip Model
Whereas the weak c oupling model deals with a situation

where two pieces of superconductor are separated by an insulating
barrier,

Josephson phe nomena have also been observed in

geometries where the barrier is not insulating.

In such situations

the so-called phas e slip process is believed to be responsible.
It is convenient to consider the case where a section of a superconducting strip is locally "weakened" by one of several techniques
(Ref.

16).

The "weakening" leads to a local decrease in the

transition temperature compared to the surroundi n g superconductor.
Often the proximity e ffect (s ee . section 1.2) is used to depress the
transition temperature in a section of the superconducting strip
and such structures are called proximity effect bridges (Ref.

If the weak link is not much longer than the.. pair d ecay length

SSN and the brid ge is operated just above the transition
ternperature of the weak section the decayin g tails of the pair
wavefunction produce a small but finite pair density within the

17).

-12 weak link (Fig.

1).

To consider the behavior of the proximity

effect bridge it will be assumed for siTnplicity that the bridge is
sufficiently narrow to be treated as one-dilnensional (Ref. 18).
In the absence of voltage the phase-slip model predicts that the
bridge behaves like an ordinary superconductor.

When voltage

is applied the lllodel predicts that a relaxation os cillation of the
supercurrent will occur.

The gradient of the pair potential

lllaintained by the battery voltage and concentrated in the weak
link region accelerates the pairs until the depairing velocity
(Ref.

19) is exceeded.

At that point the further existence of

pairs is energetically unfavorable and the pairs will depair.
(Such depairing is a non-equilibriulll dissipative process and its
occurrence invalidates the assulllptions (Ref. 20) of the weak
couplingrnodel.)

However,

the junction region,

if depairing becOlnes cOlllplete in

only normal current is conducted by the

bridge--a situation which is also energetically unfavorable.

The

pair wavefunction therefore reforms but the reformed pairs have
a subcritical velocity .

They are again accelerated and the

process repeats itself.

A rigorous treabnent of this phenolllenon

would involve the dynalllics of pairing and depairing in space and
time (cf. the weak coupling model).

A complete theory of the

phase - slip p r ocess is not currently available but attelllpts have
been made to extract the significant features.

These use the

time -dependent Ginzbur g-Landau theory (Ref. 21) or various
forllls of weak coupling theory with the inclusion of SOllle nonequilibriulll effects (Ref. 22).

The gist of the phase-slip theory

-13and some results w ill be presented.
The weak link is treated as a superposition of an S-N and
N -S boundary separated by the length of the link (Fig.

1).

zero current the pair wavefunction decays with decay length

For

S SN

(see section 1.2) from both sides towards the center of the bridge.
Since the supercurrent density is (assuming the vector potential

A = 0)

(1. 30)

there is no phase difference acros s . the bridge for zero supercurrent.

In order to change the supercurrent a voltage ·m ust be

applied across the bridge since,
ft ~

by definition
(1. 31)

2f-Lp

the total phase difference across the bridge will be

CPz - CPl =

2(f-Lp - f-Lp )

(1. 32)

The phase difference is not distributed evenly along the weak
link.

At the center where the pair density is the smallest the

flow of supercurrent must be accompanied by a large phase
gradient.

During the acceleration part of the cycle the phase

gradient increas e s until the critical value
at the center.

I. Ilep!. ~ _1

is reached
SN
In the GL theory the pair density is given by
(1. 33 )

so that when

I vepl = l/SSN the pair density is zero and supercon-

ductivity breaks do w n at the center.

This allows the phase to

slip by 2 TT and decr ea s e the phase gradient so that a finite pair

-14-

Figure 1.

(not to scale)

The distribution of pair density p along
a proximity effect bridge of length L.

-lS-

(.f)

-1

....

"-(f)

~.J.,.,P

J.i

t>O
.r<

(\J

Q..

-r-

«A

-1

-16 density can be ree stablished.

Detailed models have been

constructed to describe the time-dependence of the supercurrent
under the phase - slip conditions (Ref.

21).

According to these

models the supercurrent through the phase-slip center undergoes
a relaxation oscillation at the Josephson frequency

2eV
= -h-

The time average supercurrent density is
0.5 - 0.6 J

where J

(1. 34)

is the peak super current density (= the critical current

density).
Experimentally it is found that in a sufficiently short weak
link (proximity effect bridge) the oscillating current extends over
the entire length of the bridge L»

SSN'

even though the size of

the region over which superconductivity breaks down periodically
is expected to be only
present.

SSN.

This is not understood In detail at

semiquantitative theory was worked out for the

situation where a phase-slip center (a small weak spot? ) is
located in a homogeneous superconducting whisker or thin film
strip (Ref.

23).

In such cases experiments show that oscillating

currents extend over distances

~ 1 Of!-. around

centers in tin whiskers and strips.

the phase-slip

This distance

is thought to be the characteristic distance (eq.

(~l Of!-m

in tin)

1. 20) for the

relaxation of the quasiparticle chemical potential flQ and the pair
potential flp to each other,

i. e.,
(1.35)

-17-

where A
~ 10 fl-m in tin.
Qp

Consistently with this. equation,

the

normal current distribution can be written as
(J

-e 'Vfl- Q

(1. 36)

where Ixl is the distance from the phase-slip center and V is
the voltage across the phase-slip center between points many
distant from it (Fig.

2).

For short weak links

(S SN«

L «

the normal and supercurrents probably have little spatial variation
along the link since the S -N boundarie s at the ends of the link
function as quasiparticle mirrors (Ref.
tial decay (eq.

24) cutting off the exponen-

1.36) before a significant drop occurs.

The spatial constancy of the supercurrent and the normal
current along a short weak link justifies the use of the two-fluid
model of the weak link in which the voltage across the link is
assumed to be

where R is the normal resistance per unit width of the link,
is the total (bias) current density and J '
is calculated at the phase-slip center.
length L

the supercurrent density,

For weak links whose

substantial deviations from the simple two-fluid

model are likely to occur.

In summary, the phase-slip model of a weak link describes
the periodic breakdown of superconductivity'·at the center of the
weak link in a region of size

~ SSN

which causes the supercurrent

at the center to undergo relaxation oscillation at the Josephson

-18-

Figure 2.

The distribution of normal current I
and the pair
(flp) and quasiparticle (p'Q) chem,ical potentials as a
function of the distance x from the phase-slip center
in a homogeneous superconductor (above), and in a
short (L < )"QP)proxirnity effect bridge (below).

-19-

"':iIf-----

tSN

<1lI)

• 0

.e ... • • •

.... ..

- -,.....

-"->-.----~"""'-....,......: •

.. •

.. .. •

. ,.

"""'-

;;'Q

• ••

••

••
·0 ••

·0

e•

.....

',,0

•• 0"
• .e ••••

fLp

----~

"..

C;;SN
<;j

fLO

) •• 'e ....aH!'''''''......_ _

fLp
Figure 2

-20-

= 2eV
Ii

frequency w

In a current biased weak link the

oscillation of the supercurrent is accompanied by the counteroscillation of the quasiparticle current.

If the weak link is

shorter than the relaxation length A
both the quasiparticle and
Qp
the pair currents are spatially uniform along the weak link and
follow the dynamics at the phase slip center.

Accordingly the

time-average supercurrent density through the entire weak link
is given by equation (1.34).
Kirschman,

Notarys and Mercereau (Ref.

25) proposed that

experimental measurements on proximity effect bridge s are
consistent with

~c [ 1 + cos(2fte SVdt~

(1.38)

This waveform differs little from detailed theoretical phase - s l ip
waveforms (Ref.
1.5

21).

The Current-Voltage Characteristic of Josephson Devices
In the previous sections the weak coupling and the phase -

slip models were presented.

The voltage in both cases was given

by the equation
(1.39)
where J ,

are the total and super current densities · and 5t. is

the resistance per square of the current carrying area of the
device.

If the current is distributed uniformly across a uniform

junction area (d.

section 1. 6) then
V = R(I

- IS)

where R ~s the total resistance of the device and IT'

(1.40)

IS are the

-21total and super currents r e spectivel y.

In the preceding equations

the capacitance of the device has been neglected.

Typically,

for

. proximity effect bridges R = O. 1.fL and C < 1 pF so that the shunt
capacitive irnp edance becomes important at f

~ 10 12 Hz = 10 3 GHz.

In DC rneasurernents the tirne average voltage V is
rneasured.

It can be shown (see Chapter 3 ) that for the weak

coupling ·m odel (Fig.

3) the time average voltage is

= R~I T2 _ I c2

whereas for the phase slip ·model (Fig.

(1.41)
3)

(1 . 42 )

In both case s I

is th e critical current,

i. e.,

the maX1mum

supercurrent of the d evice at the given temperature.
cally,

for ITIIc

Asymptoti-

1 we obtain in the weak coupling model tha t
V ...

RIT

but in the phase-slip model

The term Ic/2 in the phase-slip characteristic 1S referred to as
"Excess supercurrent I I and is a direct consequence of the nonequilibrium nature of the model.

Experimentally,

R is roughl y

the resistance of the bridge at a temperature above the onset of
any measurable supercurrent •.

Excess supercurrent ~O. 5 I

seen in proximity effect bridges (Ref. 26),
w his kers and S-N-S junctions.
semiconductor barrier bridges.

tin bridges and

It is not seen in insulating or

-22-

Figure 3.

The I-V characteristics (nor"malized) of a Josephson
device of resistance R and critical current Ic in the
weak coupling model (solid curve) and in the phaseslip model (dashed curve).
The respe c tive asymptotes
are also shown (thin lines).

- 23 -

Figure 3

-24The addition of an RF current to the biasing current of a
Josephson device causes constant voltage "steps" to appear in the
I-V characteristic of the device (Fig,

4),

The mathematical

details of this behavior are complicated but have been subject to
much investigation (Ref, 27),

Physically the steps are a result

of phase locking between the external RF current and the oscillating supercurrent of the bridge,
pulling,

acco'mpanied by frequency

so that the voltage (and also the frequency of the super-

current oscillation) stays constant over a range of DC bias
currents.

The phenomenon has been used in 'microwave and far

infrared detectors.

1, 6

Quantum Interference Effects in Magnetic Fields
The phenomenon of quantum interference in magnetic fields

has been of 'much importance both in the understanding of superconductivity as a macroscopic quantum state (Ref.
applications (Ref,

28) and in

Experimentally it is manifested by

29).

periodic changes of the supercurrent as a function of magnetic
field.

Fundamentally,

the effects stern from the requirement of

gauge invariance of the supercurrent density.
The behavior of a thin film Josephson device (= bridge)
will be considered next.

Suppose that a bridge is formed in a

thin superconducting film by local weakening of the superconductor
(Fig.

5),

The structure will be described by a general phase-

supercurrent density relation (gauge invariant)

(1. 43)

-25-

Figure 4.

External RF (2 GHz) induced steps in the I-V and
dV /dI vs . I characteristics of an experimental
proximity effect bridge.
The dashed curve is the
characteristic in the absence of external RF radiation.

-2 6 -

....,...
OJ

<;j""

C\.J

e;--'

C\.J

<;j""

>10--<

Cll

'----'

"0"0

..---,

::i..
.......

C\.J

~I--------_~~------~--------~O
(!)
C\.J

>..:;t

.....~

-27-

Figure 5.

The geom.etry of a bridge of length L, width w, and
thickness d.

-28-

(W 0)
2'

( -w 0)
2'

Figure 5

-29where f attains a maxim.um. value of 1 and cp(y) is the phase
difference across the bridge between th e points (0, y) and (L, y ).
It is assumed that both the phase cp and the vector potential

A = (A x , Ay' A z ) do not vary o ver the film thickness d. (In Nb-Ta
proximity effect bridges d ~ 100 A, width w = 5-50!-lIn, L ::: 0.31.0 !-lIn).

The supercurrent variation across the thickness of the

film is therefore neglected and the sup ercurrent IS is given by

(1.44 )
Expressing the transverse dependence of the phase as

cp(y)

= J ~ dy + cp(O)

(1. 45)

and using equation (1.7)
ocp(y)/oy = oCP1 (L, y)/oy - oCP2 (0, y)/oy
ocp(y)/oy

= -2 e"p
~ [Jy (L, y)- J y (0, y)] -

:e
fI

(1.46 )
[A (L, y)- A (0,

yiJ

the equation for the bridge supercurrent can be rewritten to obtain

where

~B(Y)::: :en [f
A.d1- 2~
{[J (L,y)- J (O,Y)]dyj
r(y)
enpo

(1.47)

The quantity

it includes

the magnetic flux through th e part of the bridge between

and y

but in addition it contains tenns due to transverse currents
scre enin g the film. at the two ends of the bridge from the bridge '

-30magnetic flux

In the absence of Inagnetic field, (ji(y) = 0 and

For small magnetic fields such that (jiB (w/Z) «

(jio the super-

current is

IcIO){'[.IO)+~e ("oJ" +11:'; 'HI.IO)))}

(1.49 )

where
-Z
(jiB =

1 w/Z

(jiB (z)dz

-w/Z

The dependence of the supe rcurrent on the magnetic field B can
be solved in a closed form if a sinusoidal current-phase relation
is as sum.ed (Ref,

30),

In general magnetic fields decreas e the

maximum super current IS in a periodic fashion so that a diffraction-like pattern (Ref,

31) is seen as a function of applied

magnetic field B.
The behavior of two bridges joined in a ring (Fig,
simpler,

6) is

It will be considered in the limit that the magnetic

field is small enough so equation . (1.49) holds.

The total phase

change along a circle going through the center of the bridge s is

For path segments outside the bridges equation (1.7) holds,

~c,o = - ~
:1 - ~
.it
Zehp

i. e.,
(1 • 51 )

so that
(1.5Z)

where the prime indicates that the bridge regions are left. out of

- 31-

Figure 6.

The superconducting quantum interfero'm eter.

-32-

tI
Figure 6

-33-

The equation can be rewritten as

the path integrals.

(1. 53)

This time

= (l\cp)AB
DC

is the gauge invariant phase which is the correct argument of
the phase-current relations.

licpdj must be a multiple of 2n,

otherwise the wavefunction would not be single -valued.

So the

quantization relation is gotten as

(; 1 - (; 2 = 2nn +.0::.
crt + ~
,t; 1. crt
n ,j,;A'
2enp ~
In this equation

(1. 54)

fA. crt is the total magnetic flux through the path

of integration consisting of contributions from external sources
and fro'm the supercurrents in the ring .

Accordingly,

iJ?K

ilio

+r+where 'liE'

ipS'

iliK are fluxes due to external magnetic fields,

self-generated magnetic fields,
'li

(1.55 )

== h/2e = 2 x 10-

Wb.

comment on the term ili

and kinetic term respectively; and

At this point it is appropriate to
For the bridges

2eV
(; = -11-

(1. 56)

is the expression indicating that a voltage is needed to accelerate
the pairs and thus cause a phase change.

In fact the pairs in

the rest of the circuit also have inertia,

so that for the whole

circuit one gets
(1,57)

-34The last term simply indicates that the EMF accelerates pairs in
the nonbridge parts of the ring as we ll.

The total super cur rent

through the two bridges is (negl ecting small "s ingle bridge" terms)

= I cl f (0 1 ) +I c2 f (2 TT n + 0 1 - 2 TT

where the quantization condition (eq.

0 20

In an experiment the phase 6

current source,

(1.58)

1. 55) was used to replace
is fixed by the external

The maximulTl super current I

that can be

passed through the interferolTleter at zero v oltage is given by the
condition

= 0

(1. 59 )

The equation is in reality quite complicated because the flux
te rlTlS

~ Sand

iliK depend on 0

through their dependence on

currents passing in the ring.
To illustrate SOlTle features of the interferolTleter dynalTlics,
the simple case of a symmetric interfero'm eter is presented.
f(o)

= sin 6 and I q = I c2 = 10 ,

Let

Then

(1.60)

The diffraction terlTlS can be included in I

for this case so that

(1.61)

Temporarily the terlTlS

shall be neglected.

The maximulTl

supercurrent for the interferometer shows quantum interference:

-35iii

-E

21 o leos
(11 ,
'i' - ) I

The current is equally divided between the two bridges.
tenns

(jj

geometry.

(1. 62)

The

ips are in fact zero due to the synlmetry of the
(The asymmetric case is described in Chapter 2.)

-36References

V. · L. Ginzburg and L.
1064 (1950)

J. Bardeen, L. N.
1 08, 11 75 ( 1 957 )

R. P. Feynman, Statistical Mechanics,
W. A. Benjamin; Inc. (1972)

ibid,

P. G. De Gennes, Superconductivity of Metals and Alloys,
pp. 145-148, W. A. Benjamin, Inc. (1966)

ibid,

B. S. Deaver and W. M. Fairbank, Phys. Rev. Letters 7,
43(1961); R. Doll and M. Nabauer, Phys. Rev. Letters, 7,
51(1961)

M. L. Yu and J. E. Mercereau,
1117 (1972)

10.

T. J. Rieger, D. J. Scalapino,
Rev. Letters 27, 1787 (1971)

11.

12.

1. Giaver,

London,

pp.

and T.

Superfluids,

vol.

John Wiley & Sons (1950)

D. Landau,

Cooper,

and J.

Sov. Phys.

JETP 20,

R. Schrieffer,

Phys. Rev.

pp. 298-303

303 - 311

3-7

Josephson,

Phys.

Phys. Rev.

and J.

Letters.1

Letters 28,

E. Mercereau,

Phys.

251 (1962)

Phys. Rev. Letters 20, 12 86 (1968); C. L. Huang
Van Duzer, Appl. Phys.Lett. 25, 753 (1974)

13.

R. P. Feynman, Statistical Mechanics,
. W. A. Benjamin, Inc. (1972)

pp.

306-311

14.

P. G. Gennes, Superconductivity of Metals and Alloys,
pp. 217-225, W. A. Benjamin, Inc. (1966)

15.

F. Auracher; P. L. Richards,
B 8, 4182 (1973)

16.

H. A. Notarys and J. E. Mercereau,
1 821 (1973)

17.

Stephen K. Decker,
Technology 1975)

Ph.

18.

R. K. Kirschman,
Technology 1972)

Ph.

D. Thesis (California Institute of

and G. 1. Rochlin,

.J. Appl.

Phys. Rev.

Phys . 44,

Thesis (California Institute of

-37References (contI d)
19.

W. F. Vinen, Superconductivity (ed. by R.
1168-1234, Marcel Dekker, Inc. (1969)

20.

M. L.
1974)

21.

T. J. Rieger, D. J. Scalapino,
Rev. Letters 27, 1787 (1971)

22.

H. A. Notarys, M . L. Yu and J. E. Mercereau,
Letters 30, 743 (1 973)

Phys. Rev.

23.

W. J. Skocpol, M. R. Beasley and M.
TeITlp. Phys. ~ 145 (1974)

Tinkham,

J. of Low

24.

A. F. Andreev,

1228 (1964)

25.

R. K. Kirschman, H. A. Notarys,
Phys. Letters 34A, 209 (1971)

26.

H. A. Notarys and J. E. Mercereau,

27.

Yu,

Ph.

P. A. Richards,
36 (1973)

D. Parks),

pp.

Thesis (California Institute of Technology

and J. E. Mercereau,

Sov. Phys. JETP.!...2,

F. Auracher,

Phys.

and J. E. Mercereau,

Physica 55,

T. Van Duzer,

424 (1971)

Proc. IEEE

28.

J. E. Mercereau, Superconductivity (ed. by R.
pp. 393-421, Marcel Dekker, Inc. (19 69)

29.

S. K. Decker and J. E. Mercereau,
347 (1973)

30.

31.

H. A. Notarys and J. E. Mercereau,
1821 (1973)

Jaklevic et al"

D. Parks)

Appl, Phys.

Phys. Rev 140A,

Lett.

23,

1628 (1965)

J. Appl. Phys. 44,

-38II.

THE PHASE-CURRENT RELATION AT ZERO VOLTAGE
IN PROXIMITY EFFECT BRIDGES

2.1 · Historical Background
The phase-supercurrent relation for insulating barrier
junctions was established to be sinusoidal (Ref.
discovery of the Jos ephson effect.

l) soon after the

For thin film. bridges the

situation rem.ained confused for a long tim.e,

since in the naive

picture a bridge can reach its critical current with an alm.ost
arbitrary phase difference acros s it,
In 1970,

Baratoff et al.

(Ref.

proportional to its length.

2) pre sented a theory (based on GL

equations) according to which the phase - current relation is periodic
with a period of 2 iT but is sinusoidal only in the limit of a very
weakly super conducting link.
(Ref.

In the sam.e year Fulton and Dynes

3) investigated experim.entally the current-phase relation at

zero voltage in Anderson-Dayem. bridges.

They concluded that

the current-phase relation is "continuous,

single valued" and

"nearly sinusoidal" for critical currents smaller than 10 !-lA.
1972 Bardeen and Johnson (Ref.

4) using microscopic theory

again proposed that the phase-current relation is sinusoidal for
normal m.etal barrier junctions in the lim.it of weak coupling but
is nonsinusoidal for strongly coupled junctions.

In 1973,

the

investigation of the phase-current relation at zero voltage ln
proxim.ity effect bridges was performed in our laboratory using
a m.ethod sim.ilar to the Fulton -Dynes experim.ent.

The results

were presented at the Am.erican Physical Society m.eeting in
San Francis co,

December,

197 3.

-392.2

The Small Asymmetric Interferometer
The small asym.metric interferometer is the key element in

the Fulton-Dynes method of measuring the phase-supercurrent
relation.

In terms of the interferometer equation (1 .58 ) (Fig.

(2 . 1 )
''small '' means that the sum of the self-induced terms is small,
and "a symllletric" means that I
a first approximation,

the critical current of the interferometer

I c == IS-max will b e gotten by setting

1 == & lmax such that the

The second terTll is thereby allowed

first term is maximized.

to vary as the magnetic flux from external sources is va ried.
The modulation of the interferometer critical current by magnetic
flux is then

c == I c2 f (211n + 1 max

Neglecting g>K'

(2.2)

iJiS (kinetic and magnetic self-induced fluxes) in

the first approximation,

the modulation is proportional to the

phase-current relation with the argument

(2nn - 0lmax - 2n(g>EIg>o»

Under these conditions a measurernent of the modulation of the
critical current of a snlall asymmetric interferometer by external
magnetic flux is equivalent to 'measuring the phase-current relation.

If the phase-current relation is not periodic with a period Zrr,
the integer n will affect the lllodulation curve when it changes.
The exper irnental realization of the "slllallne s s" condition in a
strict manner,

i. eo,

iJiS + iJi

iJi

is difficult silllultaneously with

-40-

Figure 1.

The parameter

is defined as

iP = iPK + iPS + iPE - niP o.

-41- .

SUPERCURRENTS INAN

INTERF~ROMETER

QUANTIZATION CONDITION: o( -

Fi gu re 1

27Tcp

°2 = CPo

-42~

the condition I

» I

To show why,

the fluxes will be written

in the form

(2. 3)

where L '

LK are the self-inductance and the kinetic inductance

(section 1.6) respectively, and II ,1
Since g;

bridges 1 and 2.

are the currents flowing through

= 2 x 10-15 Wb,

it would be necessary

to have
(2.4)

However,

the typical current noise (Ref. 5) of proximity effect

bridges is

~ O. 1 flA

~50flA,

so that the current II

» 12 would have to be

i.e.,

+ LS «

4 x 10-llH

(2. 5)

The inductance 4 x 10 -11 H corresponds to a linear dimension
~4

x 10

-5

= 40 flm.

The dimensions of the interferometer

would have to be much smaller than 40 flm to satisfy the strict
smallness requirement,
A second approximation will therefore be used in which

o1 max'

iP~

and iPS are corrected to first order using the first

approximation.

Since it is possible to make interferometers with

a diameter of 15 II. and I

101

c2'

the second apI)roximation can

be introduced numerically after the approximate character of the
phase-current relation is known from a magnetic modulation
experiment.

-43-

2. 3

Sample Preparation
2.3. 1

General procedure

A multil ayered metal film is evaporated on an insulating
Subsequently photolitho-

wafer under ultrahigh vacuum conditions.

graphic techniques are us ed to mask selectively some areas of
the film .

In unmasked areas the film is anodized to predetermined

depth so that a l ayer of anodic oxide is forrned.
facture of proximity effect bridges (Ref.

For the manu-

6) .the top film layer of

higher intrinsic transition temperature is anodized away along a
narrow

(~ l j-1),

rectangular area leaving the bottom film layers of

lower intrinsi c transition tem.perature as the only path of conduction.

A bridge structure is thus formed where two areas of

unanodized film are joined by an area where the top layer of the
filtn is anodized away (Fi g.
rings,

2),

Other conduction paths (leads,

etc.) can be delineated by complete anodization of the film

in places where conduction is not desired.

In such films the

unanodized regions have a higher transition temper ature than the
partially anodized regions while co'm.pletely anodized regions are
insulating .
2.3.2

The substrate and the film

A film.-substrate combination for us e

= the preparation of

proximity effect bridges and ancillary structures by anodization
(Ref.

7) must meet several requirements.

', First,

the film must

show a decreasing super conducting transition te'm.perature in the
liquid helium range as a function of the depth of anodization.
Second,

the anodic oxide must be stable and insoluble in the

-44-

Figure 2.

Diagram of a proximity effect bridge.
The evaporated
Nb/Ta sandwich with thickness ts/t is anodized in the
bridge region.
The author uses brl]:dges with. t~ == 0,
t' < t and l ength t == O. 7 - 1 . 0 fJ.m.

-45-

-46reagents used during the preparation of the salllple.

Third,

film should be tough and adhere well to the substrate.
requirement,

the

The fourth

while not critical in the inte rfero-rn e ter eXperilllent,

was nevertheless found useful in other applications:

the substrate

should have a high therm.al conductivity at liquid heliulll telllperatur es to lllini"Inize the telllperature rise due to Joule heating.
The following procedure has been used (Ref.
ferollleter salllples.

An inch by inch square,

thin sapphire

substrate is cleaned by washing in chrolllerge,
reagent quality acetone successively.

8) for inter-

distilled water and

The sapphire chip is then

drie d and placed into an ultra-high vaCUUlll electron bealll
evaporato r.

The substrate is heated to 400 C and when pressure

drops to the low 10

-8

ran ge,

100-200 A of tantalum. (Ta) is

evaporated followed illlllle diately by 100-200 A of niobiulll (Nb).
The thicknes s is lllonitored by a Sloan lllonitor during evaporation.
2.3.3

Photolithography and anodization (Ref.

After evaporation the fillll is cleaned in chro'm erge,
distilled water and acetone again.

Photoresist ("PR") (Shipley AZ)

is then spun on and test holes are exposed.

The thickne s s of

the Nb and Ta layers is checked by slow anodization with a
voltage ramp.

The empirical conversion constants between

anodization voltage and film thickness anodized to oxide are:
8 A/V for Nb,
cases.

6 A/V for Ta,

yielding 15 A/V of oxide in both

After dissolving the old photoresist with acetone,

a new

layer is spun on and an interferometer rin g lllask is lllicroprojected
(in reverse) through a x lOO oil illllllersion lens.

(SOx di-rninution

-47of the pattern is achieved.)

The developed PR pattern covers a

ring structure with two leads.

Complete anodization removes all

metal film not covered by PRo leaving a thin film ring with two
The manufacture of bridges proceeds similarly.

A slit

pattern ismicroprojected through a xlOO oil immersion lens into
freshly spun PR on one of the arms of the previously made ring
pattern.

As a result,

after developing,

the PR is removed over

a strip ~ 1 fl wide extending across one arm of the interferometer
ring.

Subsequent partial anodization removes the top layer of Nb

and some Ta through the gap in the PRo
made in the other arm the same way.

The second bridge is
In order to achieve

asymmetry of roughly the desired magnitude the two bridges are
made with somewhat different anodization voltages.

A series of

interferometers is made on one substrate and the suitable ones
(I

~ 1/10) are selected through further testing.

sa'mples 'made by this procedure are shown in Fig.

Two of the
3.

These two

were chosen from a total of eight complete interferometers and
were used for all the experiments described in this chapter.
2.4

The Measurement of the Interferometer Critical Current as
a Function of Applied Magnetic Field
After the manufacture of an interferometer

completed.

two wires are attached to each of the two interferometer leads by
pressed indiu'm contacts so that a four-terminal measurement can
be perfo rmed.

The interferometer is then 'mounted inside of a

solenoid with the plane of the interferometer ring perpendicular
to the axis of the solenoid.

The solenoid is positioned at the end

CIT 13c #2 ,

Fi gur e 3

lOp.

PHOTOMICROGRAPHS OF EXPERIMENTAL

CiT 16a #6

INTERFEROMETERS

>f:>

-49of a cryogenic probe and the probe is lowered into a cryostat.
Precautions have to be taken to lilnit AC and RF interference as
well as to lllini'lllize th e effects of a'mbient lllagnetic field.

These

precautions include the placelllent of the cryostat in a shielded
roolll,

the provision of a lllagnetic shield around the cryostat,

and

the installation of a lead foil bucket inside the cryostat to create
a superconducting shield around the cryogenic probe.
An electronic feedba ck systelll is used to lllaintain the bias
current of the interferollle ter at its critical current even as the
critical current varies with the applied lllagnetic field (Fig. 4).
The operation of the feedback loop is describ e d in Fig.

The

addition of an unchopped current source consisting of the voltage
source V E and bias resistor R

(Fig. 4) decreases the necessary

offset voltage V F and thus lowers the operating point voltage V.
The operating point voltage is chosen to lie just above the high
curvature knee of the I-V characteristic so that as little distortion
of the critical current waveforlll as possible is introduced (Fig.

6).

The data are obtained in the fonn of an x-y plot with the horizontal
axis x proportional to the solenoid current IB and thus proportional
to the Il".agnetic flux through the interferollleter due to the solenoid.
The vertical axis y is driven by the output of the lock-in alllplifier
(including its offset voltage V F) and is therefore proportional to
the critical current I c of the interferolneter, . (Fig.

4).

Since the

critical currents of the bridges are a function of the telllperature
of the bath,

Ic vs. IB plots are recorded at several telllperatures.

-50-

DIAGRAM OF THE CIRCUIT FOR MEASURING Ie

I ..

. - - - - - < : , - - - - - 1 Sy~~ C.

r-----.. CHOPPF"R
REt--.

""'"" Vout
GAIN =G

Figure 4

. TO CHART
RECORDER

-51-

THE WORKIf\JG POINT OF THE CIRCUIT FOR
MEASURING Ie (TYPICAL VALUES)

JUNCTION
V{ I) = CHAR:~CTERISTIC

V [nV]

. Figure 5

-52-

Fi gure 6.

(experimental tr ac in g ) The choi ce of operating point
of the circuit in Figures 4 and 5.
Themagnetic
fields Bmax and Bmin produc e the minima and maxima
of the int erference modulation of the c ritical current
I .

-5 3 -

'U>'
+0

(!)

L-.:..I

=#: 00

(!)

r<>

l- I(.)

r<>

to
II

ll..

...

...---,

:t.

L--.J

.....~

...0
Ql

.r<

-542.5

Data Analysis,

Results and Discussion

Two interferolneters (Fig. 3) were used to obtain the data
to be presented.
asymlTIetry I

o< I

They were selected because in both cases the
cl

was roughly 1 :10 in the current range

< 50 f.LA so that a cOlTIprolnise between the requirelTIents

of slTIallness and asyrnlTIetry could be achieved.
Initial lTIeasurelTIents indicated that the lTIodulation was
roughly sinusoidal on a slowly varying background due to single
bridge diffraction (Fig.

7).

No effects attributable to the

periodicity of the phase-current relation differing frOlTI 2n were
seen (section 2.2).

At this point the approach was reversed.

The working hypothesis becalTIe that the phase-current relationship
is sinusoidal with a period 2n,

and an attelTIpt was lTIade to detect

any deviations incolTIpatible with this hypothesis.

There are three

causes of the int erferolTIeter critical current lTIodulation deviating
frolTI the sinusoid that are cOlTIpatible with a · sinusoidal currentphase relation.

The first of these is trivial:

the single junction

diffraction terlTI causes the overall background curvature of the
lTIodulated waveform.

For this reason it was decided to · analyze

only the few lTIodulation periods near the peak of the background
waveforlTI where the variation is the slowest.

The second effect

is due to the inductance of the interferolTIeter ring L

= LK + L S '

the sum of the kinetic and magnetic self-inductances (s ee section
2.3).

As a re sult of kinetic and lTIagnetic flux terlTIS due to

currents flowing in the ring,

the central peak of the lTIodulation

pattern is shifted away frolTI the point where the external lTIagnetic

-100 -80

I-~~

-60

-40

-20

Figure 7

100

t1>

Bext [mAl

-~J

CIT 13e #2
T= 4.16°K

Ie vs. B IN AN INTERFEROMETER WITH ICI = 12 IC2
Ie [f-L AJ
lJl
lJl

-56field is zero,
(Fig.

8).

and a tilt is introduced into the modulation pattern

The tilt is caused by the changes in 12 (the supercurrent

through the weaker bridge) as the external magnetic field is varied,
When 12 flows in one direction it increases the total flux through
the ring,

in turn causing a faster rise of the current 12 as a

function of external magnetic field.

"\Vhen 12 reverses it produces

a flux term which diminishes the total flux causing a slower fall
Both . a shift and a

in 12 as a function of external magnetic field.

tilt were observed in experiments with the two interferometers.
The shift should vary linearly with I
proportional to I

constants L' = L.

while the tilt should be

with approximately the same proportionality
Experimentally,

the. case (Table 1).

Finally,

this was indeed found to be

the last compatible caus e of the

modulation deviating from the sinusoid is due to imperfect
asymmetry I

»1

With the use of trigonometric identities

the interferoITleter equation (2. 1) can be rewritten to show that the
lowest order correction (for a sinusoidal phase-current relation)
to equation (2.2) is

I 2
ill (1) =

(2. 6)

-2 -1-

This correction has a constant component and a component at the
second harmonic of the sinusoidal modulation pattern.
tude of the second harmonic correction is
Accordingly,

the data,

2/41

recorded ln the form of Ic vs. 1B

plots as described in section 2.4,
fashion.

-I

The ampli-

were analyzed in the following

The curves were fitted with the analytical form

-57-

Figure 8.

(Sketch)

The effect of inductance L of the interferometer
on the modulated waveform.

- 58 -

THE EFFECT OF INDUCTANCE ON Ie
IN INTERFEROMETERS WITH lei» Ie2 .

Fi gure 8

-59( d.

equation Z. I)

= I

Sln (~

- Zl1 '!!_gi )

. CT

l - - LI Cz Sln-Z
IB == M

(Z. 7)

~,l
)-

Zrr :

where iji is a free parameter corresponding to the total flux through
the interferometer and where I
the two bridges,

the flux quantum,

cz'

are the critical currents of
gio is

MIB is the flux through the interferometer due

the background B field.
cl

is the inductance of the interferometer,

to the solenoid current I ,

constants I

and iji' is the flux due to II = I

and

It should be noted that all of the
iji',

M are gotten by fitting.

Internal

checks of the inductance L can be made by comparing data
recorded at various telnperatures (i. e.,

various critical currents

In addition an independent measurement of L' :, L

can be made from the shift of the central maximum with changing

as the temperature is varied (i. e.,

L' '=

[H'/aI

A priori,

it would seem that the mutual inductance M of the solenoid and
interferometer in equation (Z.7) can be calculated from the
geometry.

However,

the interferometer is in close proximity to

lar ge contact pads of super conducting film.

Since the supercon-

ductor is strongly diamagnetic the magnetic flux expelled froIll the
pads is concentrated in the interferoIlleter.

The mutual inductance

M therefore turns out to be Illuch lar ger than M

geoIll

calculated

neglecting diaIllagnetic effects (Table I).
Finally,

the deviation of the data froIll the fit (equation Z. 7),

-60if significant,

was compared to that expected from imperfect

asymmetry (equation 2.6).
The results are presented ln Table 1 and in Figures 9 and

10.

For the four curves analyzed in detail the noise amplitudes

N were from 2 to 8 per cent of the lTIodulation amplitude I

and
c2
it is estimated that an incompatible periodic deviation would be
detected if its amplitude exceeded 5 percent of the modulation
amplitude I

No such deviation was found in any of the data

studied.
It should be noted that this study was done on bridges at

zero voltage.

Due to the onset of nonequilibrium behavior at

finite voltages extrapolation of the zero voltage phase-current
relation to finite voltages is not warranted.

Additionally,

the

phase-current relation is thought to depend on the strength of
coupling between the two superconductors separated by the normal
barrier,

i. e.,

on the geometry and material composition of the

bridge.

The "coupling strength" of the theoretical models is

closely related to supercurrent density.

In this study the super-

current density is estimated to be about 50-100 iJ-A/iJ-m2 which is
of the same order of magnitude as the super current density in
the typical working regime of most Nb/Ta proximity effect bridges
but about ten time s higher than the current density in insulating
barrier junctions.

It is likely that significant deviations from the

sinusoidal phase-current relation at zero voltage will not be
observed unless m .uch higher supercurrent densities are reached.

Fit:

L!~

c alculated using L == 1.5x IO-lI H

Z. 6 .

2. 6
0.19

0 . 08

O.ost

O. l Z

O. OS ':'

O.OZ

O. OZ

IC2

O. OZ

0 . 04

0.08

Curve used only for measurement of Lan d L '

1.9

1. 5

Z IcZL
IT ~

3.30

1.8Z

1. 04

1. 34

0 .6 1

[lO-llH] [lO-llH]

L " 1. 5 x 10-li H u s ed m fitting . Second ha r moni c deviation detected but it agrees with equation 2 . 6

39.75

ZZ . 13

cal culated u sing L " 3.x I O- II H

1Z. 3Z

[f-LA]

ZITI

ZITI C L!~ 0

3. 7

11 . 04

6. 6

6.65

6. 6
2.4

[f-LA)

[10 - l3 H ]

[I O- I3 H]

Mgeom

Fits within noise w ith L " 0 in equation 2 . 7
II
Fits within nois e with L " 3 x 10- H

tIndex of tilt

~'Index of t ilt

[f-Lm]

Width

R",0 . 4J\.

CIT 13c ';;2

R " 1. 6J\.

CIT 16a #6

[f-Lm]

Mean
Diameter

Interferome t e r

TABLE

Fit

>-'

-62-

Figure 9.

Comparison of an experimental trace with theory
(equation 2.7)

-5

-4

CIT 13c #2
T=4.IOOr(

Figure 9

· Ic [f-LAJ

THEORY

B [mAl

-64-

Figure 10 ;

Comparison of an experimental trace with theory
(equation 2. 7)

-65-

0::

I-

ill
-I'

"t~

.-I

(!)

ill

'-'

....So

r<>

I-

,.----,

::L

If)

(\J

L--J

If)

(»

-662. 6

Conclusion
The phase-supercurrent relation in proxi-m ity effect bridges

at zero voltage was experimentally determined using two asymITletric quantum interferometers.

With supercurrent density in the

weaker bridge estiITlated at 50-100 I-LA/I-Lm

no evidence of

deviation froITl the Josephson phase-supercurrent relation IS = Icsinli
was found.

Theexperi-m ent was sufficiently sensitive to detect

deviations as small as 0.05 I .
From theory (Refs.

2 and 4) it is expected that the

deviation froITl the Josephson relation increases with the "strength
of coupling",

i. e.,

the supercurrent density in the bridge.

The

supercurrent density in this experiITlent is typical of proximity
effect bridges ln general but is about an order of -m agnitude
larger than themaximu-m in insulating barrier Josephson junctions.
Until any future evidence shows otherwise the siITlple Josephson
relation can be used to describe the zero voltage regiITle of both
proxiITlity effect bridges and insulating barrier Josephson junctions.

-67References

Lfaklevic et al.,

A. Baratoff, J. A. Blackburn and B.
Rev. Letters 25, 1096 (1970)

T. A. Fulton and R.
794 (1970)

Dynes,

Phys. Rev.

Letters 25,

Johnson,

Phys. Rev .

5B,

R. K. Kirschman and J. E. Mercereau,
177 (1971)

H. A. Notarys and J. E. Mercereau,
1821 (1973)

D. W. Palmer and S. K.
1621 (1973)

S. K. Decke r, Ph.
Technology 1975)

Bardeen and J.

Phys. Rev.

Decker,

140A,

1628 (1965)

B. Schwartz,

Phys.

72 (1972)

Phys.

Lett.

35A,

J. Appl.

Phys.

44~

Rev. Sci. Instrum. 44,

Thesis (California Institute of

-68III.

THE INTERACTION OF PROXIMITY EFFECT BRIDGES
WITH SUPER CONDUCTING MICROSTRIP RESONATORS

3. 1

Introduction
When Josephson devices are made to interact with RF

resonato rs,

the I -V characteristics of the device s (s ection 1. 5)

are modified.

Usually,

near the voltage corresponding to the

resonant frequency of the cavity,
the I-V characteristic.

a step-like structure appears in

To distinguish the structure from similar

"steps" induced by external RF currents (see section 1.5) the
resonator caused features are often referred to as "self-induced
Their study is of interest both for the characterization

steps".

of Josephson devices and for device applications (Ref.

1).

The

first experimental observation of self-induced steps was reported
by Fiske (Ref.

2) for insulating barrier Josephson junction

interacting with stripline type modes within the junction itself.
Subsequently self-induced steps in the I-V characteristics of
point-contact devices placed in a cylindrical cavity were observed
by Dayem and Grimes (Ref.

3).

In 1974 L evinsen (Ref. 4) saw

self-induced steps with a Dayem bridge coupled to a rectangular
microwave cavity.

Since thin film bridges are planar devices it

was decided in this laboratory to use "planar" resonators--mi crostrip resonators--for the exploration of the interaction of proximity
effect bridges with RF resonant systems.

A preliminary report

on these studies will appear in the Applied Physics Journal
(Ref.

5).
Several goals have been pursued In this work.

Initially,

-69the technology of microcircuits with microstrips and proximity
effect bridges was developed .

Secondly,

the phenomenon of

self-induce d steps was studied experimentally in these microcircuits and qualitatively compared to simple models.

Finally,

. an attempt was ·made to compare quantitatively the size and shape
of the steps observed in the experiments to those predicted from
two alternative theories of bridge dynamics and estimate the
amplitude of bridge oscillation in the GHz range.
3.2

Self-induced Steps --General Considerations
According to the two-fluid model (Ref.

6),

a thin film

bridge can be considered as consisting of an ideal "junction
element " and a shunting re sistor.

The junction element is a

voltage controlled oscillator which allows the flow of a supercurrent
(3. l)

where f is a periodic function with period 2TT and V is the voltage
across the bridge.

In the current-biased mode the shunting

resistor R carries the normal part of the bias current I so that
the voltage across the bridge is

(3. 2)

The voltage measured in the I-V characteristic of bridge s is the
time average of equation (3.2),

i.e.,

In this model all the deviations from a simple resistive
cha racteris tic

(3. 3)

-70-

= RI

(3.4)

are ascribed to the time average of the supercurrent IS'

Two

factors determine the luagnitude of the average supercurrent IS.
The first is the intrinsic dynamics of the bridge,
by the wavefonu f

in equation (3. 1).

here represented

The second factor is

self-modulation due to the oscillating behavior of the timedependent voltage V,

caused by the oscillations in the current

flowing through the shunting resistor R.

The voltage V oscillates

about its average V thus alternately speeding the rate of phase
development when the supercurrent IS is negative (= opposite to
the bias current I) and slowing the rate down when the supercurrent IS is positive.

This effect by itself increases the average

super current IS beyond that given by the phase average of the
phase-supercurrent relation IS = Ic f(o).

When a bridge is

strongly coupled to a resonator an additional shunting impedance
Z(w) is added to the bridge circuit (Fig.

1) to account for the

part of the bias current flowing through the resonator.

Assuming

that the real part of the resonator impedance is negligible (or
more accurately,

that the real part of the resonator adlnittance

iSlnuch smaller than l/R) the alnount of self-modulation will
change as the oscillation frequency of the bridge passes through
the resonance of the shunting resonator.
There are two silnple cases of inter~st.

In the first case

the 'lnagnitude of the ilnpedance Z(w) is luuch greater than the
shunt resistance R except near resonance (0

o where Z(w 0 ) «R.

A ccordingly the I-V characteristic reflects the self-modulation

-71-

Figure 10

The equivalent circuit of a proximity effect bridge of
resistance R shunted by a resonator.

-72-

Figure 1

-73appropriate to the resistance R (as if the resonator were absent)
except when the fundamental oscillation frequency of the bridge is
near the resonant frequency w ,

On resonance the shunting

resistor R is shorted by the resonator at the fundamental frequency
of the bridge oscillation and self-modulation almost ceases,

(The

higher harmonics may still contribute a small amount of selfmodulation.)

As a result the time average supercurrent IS is

lower at resonance than it would be in the absence of the
resonator.

The time average voltage V is thus higher at

resonance producing an upward (convex) step in the I-V characteristic (Fig,

2B),

In the second case the resonator impedance Z(w)

is much smaller than the shunt resistor R except near the
resonant frequency wo'
case,)

(The situation is the reverse of the first

The self-modulation is very small except when the bridge

oscillates at the resonator frequency,

As a result the time

average voltage V is higher than it would be in the absence of a
resonator except on resonance where it drops roughly to what it
would be in the absence of a resonator,
step is thus produced (Fig,

2A).

A downward (concave)

In both kinds of resonator-bridge

circuits the maximum size of the self-induced step 6V is the
voltage difference at a given bias current I between the characteristics of the bridge V(I) with and without the effect of selfmodulation,
This introductory discussion
aspects,

oversimplified in several

The effects of noise have been ignored,

Additional

complications stern from the multiple valuedness of the time-

-74-

Figure 2.

Sketch of a self-induced step in the I-V characteristic
of a bridge (IS =0 Ic sin 0) of resistance R coupled to
a transrnis sion line of characteristic impe dance Zo.
In A:
Zo ~ RIB, while in B:
Zo ~ SR.
Dotted line
is the interpolated nonresonant characteristic.
Dashed
line is the characteristic in the absence of a resonator.
(Based on co'r nputer simulation.)

-75-

V,
RIc

=*t r---~

- / .

n=1

RIc~--------~~/~-----X/

RIc

-.---------------

Rlc~--------~~~--

Fi gure 2

-76average voltage Vii) as a function of the bias current for a
certain "range of bridge and resonator paraITleters and fr o-rn the
possibly anharITlonic nature of the phase-supercurrent relation
These topics will be discussed in section 3.5 where
a ITlore detailed treatITlent will be presented.
3.3

ExperiITlental Technique
3 . 3.1

The bridge-resonator circuit

The coupled bridge-resonator circuits were constructed on
single silicon or sapphire chips using super conducting ITlicrocircuitry techniques (Ref.

7).

Four types of resonator circuits

were ITlade (coded 1-4).

In the type 1 (Fig.

3) a two-layer

niobiuITl on tantaluITl (Nb/Ta = 280 A/260A) filITl was deposited
(see section 2.3) on a sapphire chip to forITl a ground plane.
dielectric strip (typ.

l5ITlITl x 2ITl"m ) was subsequently forITled on

the ground plane by controlled anodization of the deposited film
to the depth of 30 V (equivalent to approxiITlately 150 A of Nb
converted into 450 A of Nb 0 ) through a photoresist pattern (see
2 5
section 2.3).

At one end of the dielectric strip a proxiITlity

effect bridge (width l5f-l x length If-L) was ITlade

by further

anodization to the depth of 65 V through another photoresist pattern.
This was followed by the forITlation of a contact pad to one side
of the bridge using cOITlplete anodization of the film to delineate
the pad.

Then a 600 f-lITl wide top strip of i200 A of Indalloy 11

was evaporated across the bridge and onto the dielectric strip.
The contact between the top strip and the ground plane is superconducting at one end of the bridge while at the other end the

-77-

Figure 3

(not to scale)

The type 1 bridge-resonator circuit.
1 = sapphire substrate. 2 = Ta film
(260 A), 3 = Nb film (280 A), 4 = anodic
Nb 2 0 5 (~450 A), 5 = Indalloy 11 top
strip, 6 = contact area: top strip to
ground plane, 7 = proximity effect
bridge.

- 78-

10 mm

Fi gure 3

-79anodized dielectric separates the ground plane from the top
Indalloy layer fonning a microstrip structure open at one end
terminated by a bridge at the other end.

Microstrips similar to

the ones e=ployed in the type 1 circuit,

with the thickness d of

the dielectric comparable to the superconducting penetration depth,
were studied in detail by Mason and Gould (Ref.

8) according to

whom the characteristic impedance of these microstrips is given
by

r;;v/c

where d is the thickness of the dielectric layer,
the microstrip,

w is the width of

v is the phase velocity and r;; is the dielectric

constant of the dielectric layer.
constant (r;; = 8-40, Ref.

As a result of the high dielectric

9) and the inductive loading of the micro-

strip by the superconductor,
microstrips.

(3.5)

vi c «

1 are measured in these

Unfortunately the intrinsic Q of these super conducting

microstrip resonators depends on the detailed properties of the
materials used and may vary significantly from sample to sample
(Ref.

8).

In the circuits used in this study the effect of the

intrinsic Q on self-induced steps was minimized by the strong
loading of the resonators by the resistance of the bridge (typically
loaded Q ~ 10 is aimed for),

Due to technological limitations the

type 1 (anodized dielectric) microstrip is best suited for characteristic impedances of 50 rnA or less,

while the typical bridge

resistance is 0.1 - 0.2A.
In the circuits of the second type (Fig, 4) the anodized

-80-

Figure 4.

(not to scale) The type 2 bridge-resonator circuit.
1 = sapphire substrate, 2 = Nb/Ta filrn,
3 = proximity effect bridge, 4 = germ.anium
dielectric, 5 = Indalloy 11 top strip. The
several samples had microstrips of width
0.1-1 mIll, length 10-15mm and dielectric
thickness 0.5-1 f.l.m.

-81-

Figure 4

-82dielectric was replaced by an evaporated high resistivity germanium
layer (typically;$ 1 f.Lm thick).
usual procedure (Ref.

First a bridge was made by the

10) in a NblTa film.

strip was evaporated across the bridge.

Next the germanium

Finally a strip of

Indalloy 11 (width 100 f.Lm to 1 mm) was evaporated onto the
germanium strip.

The length of the top strip was set by the

removal of unwanted Indalloy by a combination of photolithography
and chemical etching.

It should be noted that no contact is

desired between the top Indalloy strip and the ground plane
containing the bridge.

The resulting structure is a microstrip

open at both ends containing a proximity effect bridge in the
ground plane at the center of the microstrip segment.

The type

2 circuits are most suitable for characteristic impedance Z
the 0.1-,,- - 1-,,-

range.

The top Indalloy strip can be selectively

che'm ically removed and reevaporated making it possible to vary
the characteristic impedance of the microstrip while retaining the
same bridge.
The third type of circuit (Fig.

5) used a high resistivity
On one side of the

silicon chip 0.4 mm thick as the dielectric.

chip the top strip containing the bridge at the center was made by
a combination of photolithography,
techniques (Ref.

anodization and plasr.na etching

11) in a Nb/Ta film (typ.

120 A/250 A).

sa'me time the bridge leads were also made.

At the

The lead geometry

was chosen to minimize the loading of the circuit by the leads at

resonance .

A ground plane film of several thousand A of Indalloy

11 was evaporated on the other face of the chip.

This techl1.ique

-83-

Figure 5.

The type 3 bridge-resonator circuit. 1 = silicon or
sapphire substrate, 2 = Nb/Ta filrn, 3 =- proximity effect
bridge, 4 = contact pads for bias (I) andnlOnitor (V) leads.
The reverse side of the substrate is covered by an
1ndalloy 11 fihn forming the ground plane .

- 84 -

29mm

8mm

Figure 5

-85is ITlost suitable for ITlicrostrips of characteristic iITlpedance Z

greater than 5-,,-.
Finally the type 4 cir cuit (Fig.

6) was constructed to study

the resonant interaction of two proxiITlity effect bridges in a
ITlicrostrip resonator.

The circuit was ITlade on a sapphire chip

0.25ITlITl thick in a Nb/Ta filITl (72 A/256 A) by the technique used
for type 3 circuits ,

The two bridges were separated by 3ITlm,

and three superconducting bridge leads were eITlployed for independent biasing and ITlonitoring of the two bridges.

A s before,

the

ground plane on the reverse side of the chip was an evaporated
filITl of Indalloy 11 (1000 A).
3,3.2

The ITleasureITlent of dV/dI vs. I and

V vs,

I characteristics

In theoretical studies it is custoITlary to work with the

V vs, I characteristic of bridges due to the convenience of
calculation,

The dV /dI vs, I characteristic was preferred

experiITlentally since it was easier to measure and gave better
resolution of s'mall features.

It was obtained by adding a sITlall

AC cOITlponent (i = 0.1 f!A RMS) to the DC bias current I
of the bridge and synchronously ITleasuring the voltage
across the bridge with a lock-in aITlplifier (HR-8,

Princeton

Applied Research) as a function of the bias current 1.
cases where a V vs, I characteristic was desired,

In those

the whole bias

current I was chopped by a synchronous chopper and the voltage
V was ITleasured by the lock-in aITlplifier across the bridge.
all cases a four-terminal ITleasureITlent was e'm ployed.

- 86-

Figure 6.

The type 4 bridge -resonator circuit.
There are two
bridges in the ITlicrostrip.
They can be biased and
ulonitored independently.
The construction of the
circuit is otherwise sim.ilar to that of type 3 circuits.

- 87-

32mm ·

6.3mm

--. --

Fi gur e 6

-88For all proximity effect bridges

dv/dI vs. I traces were

obtained both before and after their inclusion in a resonant circuit,
Incr easing and decreasing bias current sweeps were used to detect
possible hysteresis.

Both bias current directions were tested in

several samples to guard against possible offset in the bias
current.
To minirnize RF interference and stray magnetic fields all
the measurements were performed in a shielded room in a cryostat
jacketed by a magnetic shield.

The tip of the cryogenic probe,

where the bridge-resonator sa'mple was located,
a superconducting lead shield.

was shielded by

To reduce the noise input through

the bias (I) and monitor (dV /dI) leads to the bridge,
were used in the cryogenic probe,

coaxial cables

In addition a lK1'l resistor

was placed into one bridge bias lead in the cryogenic space.

transformer input (type B) prea'm plifier was used in all experiments.
3 ,4

Observations
3.4. 1

Self-induced steps

Experimentally,

"self-induced steps" in the V vs, I

characteristic of a bridge coupled to a resonant system are steplike features satisfying three crite ria:
a) the steps occur in the absence of ex te rnal RF signal
only when the bridge is coupled to the resonant system,
b) the steps are at voltages (frequencies) corresponding to
the modes of the resonant system,

and

c) no steps are seen above the transition temperature of
the bridge (i , e"

above the temperature where the bridge

-89 begins to carry a detectable supercurrent).
Self-induced steps satisfying these criteria were seen in the I-V
characteristics of bridge-resonator circuits of all four types.
Usually the first derivative of such steps was recorded in the
dV jdI vs. I characteristic of these circuits where a convex step
showed up as a crest followed by a trough whereas a concave
step appeared as a trough followed by a crest.
The circuits of the first type contained a microstrip
resonator which acted as an RF short off resonance but had a
high impedance on resonance (relative to the bridge resistance R).
These circuits yielded concave steps (Figs.

7 and 8).

On the

other hand the circuits of types 2-4 contained resonators with
relatively high impedance off resonance but their impedance on
resonance acted as an RF short for the bridge.
gave convex steps (Figs.

9 and 10).

These circuits

In all cases the knee of the

step was at a voltage (frequency) corresponding to a mode of
the resonator as well as could be determined by a priori calculations.

Commonly several steps could be observed corresponding

to the sequential mode s of the re sonator.

The fundamental

resonances were in the 0.6 GHz - 4 GHz range depending on the
length and composition of the microstrip.

The size of the steps

was a function of the critical current of the bridge (i. e.,

of the

temperature of the bath) with steps beco'm ing more prominent at
higher critical currents (Fig.

10).

When several steps were

present their amplitude would decrease with increasing mode
frequency until they beca'me unobservable (usually above 10 GHz).

- 90 -

Figure 7.

The characteristics of a type 1 bridge-resonator
circuit (AF-l).
The resonant frequencies are
sequential multiples of the lowest resonant frequency
fl
O. 7 GHz.
Voltages V corresponding to the
re sonant frequencies are indicated by arrows.
The
characteristic impedance of the micro strip is
estimated as Z = (30±Z O)rn.n. (depending on the
as sumedmicro~trip dielectric constant E:), while
the bridge re sistance is R == 135 'mn.

- 91-

1000'

800

' 600

dV/dI
(mn)

400

200

' 0

____ ____ ____ ____

- L_ _ _ _~

4.44

(/-LV)

2.22

I (fLA)
F i gure 7

-92-

Figure 8.

The dV jdI vs. I characteristic of a bridge (AF-2)
before (top graph) and after (bottom graph) its
inclusion in a type 1 bridge-resonator circuit
(Z :::: 30±20 m.", R :::: 130 m.n.).
Both traces were
recorded at the same temperature of the bath.

- 93 -

300

dV/dI
(mil)

200

100

...

300

dV/dI
(mil)

200

100

. I (fLA)
F igure 8

-94 -

Figure 9.

The dV jdI vs. I characteristics of a type 2 circuit
(CIT 16 BC) at two different bath temperatures.
The most pro"m inent step occurring at a bias
current 40-55 flA is due to the interaction of tJ:-:e
Josephson oscillation with the lowest resonant
frequency fl == 1.9 GHz.
The smaller steps are
due to the second harmonic of the Josephson
oscillation.

-9 5-

dV/dI
(mil)
100

100

dV/dI
(mn)

100

O .

I (fLA)
Figur e 9

100

-96-

Figure 10.

The dV jdJ. vs. I characteristics of a bridge (0-1)
before (top graph) and after(bottom graph) its
inclusion in a type 3 resonant circuit (Z = 5.5J·"
R := 170 m .l1. ).
Both graphs were recorded at
th e same temperature.

-97-

dV/dI
(mn)

200

100

o ~-----bl~__~~____~__~~____~
10
20
30
40
50
300

dV/dI
(mn) .

200

100

I (/LA)
Fi gure 10

-98In circuits 2-4,

steps corresponding to the interaction of the

second harmonic of the bridge oscillation with the resonant modes
were also seen but their amplitude was relatively small (Fig.

10).

It was noted that the resonators with a lower loaded Q yielded

relatively broader steps in the bias current domain than those
with a higher loaded Q (d. Figs.

9 and 10).

The type 3 circuits with the highe st Q ( Q = 40 at 1. 56 GHz)
were seen to be very sensitive to ambient RF interference.

When

the door of the screen roo'm was opened the step size would
decrease and the steps would broaden.
with circuits of lower Q.

No such effect was seen

To account for these observations ,a

more detailed model of the interaction of proximity effe ct bridge s
with microstrip resonators was developed (section 3.5).
3.4.2

Resonant interaction between two bridges coupled
by a microstrip resonator

Preliminary observations were made in one circuit of
type 4.

The circuit contained two bridges approximately 3 mm

apart at the center of the top strip of a microstrip segment
(Fig.

6).

The two bridges were biased and monitored indepen-

dently of each other.

The dV /dI vs. I characteristics of both

bridges showed a self-induced step corresponding to the lowest
resonant mode of the microstrip resonator.

The size and shape

of the self-induced step of one bridge changed markedly depending
on the bias point of the other bridge (Fig.

11).

Qualitatively,

the behavior of the system can be understood in terms of two
effects:

first,

the contribution of the impedance of the second

-99-

Figure 11.

The characteristics of two coupled bridges in the
type 4 circuit.
Bridge 1 characteristics (graphs
A, B, C) are given as a function of bridge 2 bias
current (graph D).
Bias points A, B, C correspond
to graphs A, B, C respectively.
For bias below the
critical current of bridge 2 (A) maximum Q is
obtained.
For step bias (B) the deepening of th e
step in graph B is evidence of phase -locking between
the bridges.
For bias (C) above the step the Q of
the s tep of bridge 1 is lower than in (A) .

-100-

~r--'"

....--..

...........

>--l

.~ 300r

-0

O·

250

150 100
50
00

20 40 60 80 1000

40 60 80 100

I [f-LAJ

I [fLAJ
Figure 11

-101bridge to the total impedance at the terminals of the first bridge;
and second.

the phase loc k ing between the two bridges when both

os c illate at the resonant frequency of the resonator.

Accordingly

the characteristic of one bridge displays a self-induced step
corresponding to a resonant mode whose Q (and possibly frequency?)
depends on the bias point of the other bridge.

When both bridges

are on resonance both di s play a characteristic which is a superposition of a self-induced step with an externally induced step.
The system consisting of two bridges coupled by a
resonator is a rich (Ref.
Its detailed exploration.

11) and mathematically complex system.
however,

is outside the scope of this

work.
3 .5

Self-induced Steps for a Simple Harmonic Phase-Supercurrent
Relation -,- Theory
3.5.1 The model equations
Two kinds of phas 'e -supercurrent relations are currently

used to describe Josephson devices (see Chapter 1).

For devices

in which the supercurrent flow at finite voltages involves tunneling
through a barrier or an equivalent process the phase-supercurrent
relation is (Ref.

12)
= I

sin Ii

(3. 6)

wher ea s for devices in which a phase-slip process occurs the
phase-supercurrent relation is believed to be (Ref . 13)

(1 + cos Ii)
= ~
At finite voltages an additional so-called "quasiparticle

(3. 7)

-102interference" term is thought to playa role in tunneling devices
(Ref.

12) .

In proxiTIlity effect bridges quasiparticle interference

effects have not been observed to date (Ref.
furth e r

14) and will not be

It should be noted that regardless of

considered here.

the phase-supercurrent relation the phase develops according to
th e relation (Chapter 1)

o =

( 3. 8)

The two phase-supercurrent ·relations equations (3.6) and (3.7)
can for siTIlplicity of writing be condensed for finite voltages as:
( 3. 9)

where q

= 1 is equivalent to equations (3.6) and ( 3.8) while

= ~ is equivalent to equation (3.7) and (3.8) .
AssuTIling the two fluid TIlodel for a bridge of resistance R

biased by a current source I,

one obtains the integral equation

(3.1 0)
which can be solved to give
V(t)

1 - a
= R [ I - (l-q)I c ] 1 +a sino t

where
a - I - (l-q)I

and

2eV'{ti
i1

The bridge voltage V(t) can be expanded in a Fourier series
V(t)
w h e re

= V

- Vlsinw t

- V cos2w t + • . .

0 0 0

(3.11)

-104-

qX(w )1

sin 1

qX(Zw )1

(3.14)

3.5.2

Step size

Equations (3.12) and (3.14) can be used to calculate the
size of the self-induced steps in the harmonic model by deter -"
mining the voltage difference to V
between the voltage V
the voltage V

"max

at a given bias current I

== V with the resonator on resonance and

with the resonator off resonance.

the size of the steps tov

max

(In experiments

is defined as the maximum voltage

difference between the experimental I-V characteristic in the
step region and the curve interpolated from outside the step
region. )

-105 As a first approximation the size of the self-induced step
is obtained by calculating the voltage difference at a gi ven current
between the situation where the resonator reactance X i s zero
and infinite respectively.

The simpl est case occurs when all the

harmonics can resonate simultaneously. (e. g ••
microstrip r esonators).
!J. Y

in certain types of

The size of the self-induced step i s then

= Y 0 (I. X (w0 ) = X ( Zw 0 ) =

max

- Y

(I, X (w )

= 0)

= X(Zw 0 ) = .•. = "" )

(3. 1 5 )

T he first and second terms are t h e time ave r aged voltages in the
absence and in t h e presence of self - modulation,

respectively .

Using equations (3.10) and (3. 12) one obtains for this case
!J.Y

max
qRI

(3 . 16)

In the second case of int eres t onl y t he funda·m ental frequency
resonat es b ut t he second harmoni c interacts with a non r eson ant
l arge reactan ce.

The size of the step is calculated as

(3.17)

and the effect of higher harmonics is neglected.

T he set of

equations (3. 14) can be so l ved by s u ccessive approximation to
show that
!J.Y

max
RI o
q c

1 3
= 2 a + 8" a + 0(a )

(3 . 1 8)

e . , to the order a 4 the res ul t is the same as for all the

harmonics resonating.

If X(w)

R for w near 2w

one obtains

-106-

(3. 19)
- V (1. X(w ) = co,

X(Zw ) = 0)

i. e.,

2 a + O(a )

(3. 20)

The last case of interest occurs when only the second harmonic
resonates

X(Zw )

= co)

(3.21)

which gives
6V

max
qRI

(3.22)

It can be seen that the second harmonic will resonate when the

Josephson frequency Wo

= 2e V/0)

is one half of the resonator

Inode frequency and the size of the second harmonic self-induced

step is smaller by a factor of ~ 4" a

than the corresponding

fundamental step would be.
3.5.3

Step shape

The shape of the self-induced step in the harmonic model
1S

considered next.

Due to the complexity of the equation (3. 14)

only the lowest order terms exhibitin g a self -induced step are
calculated.

Accordingly equations (3.14) are approximated as

qRI

cos 6

-107-

V ·
qX(w )1
o c

sinal

(3 . 23 )

In this approximation the time average voltage V

qRI

= a

i s gi ven by

'2 a

(3. 24 )

1 + R2 j X2 (w )

If the functional form of X(w ) == X (2eV jh)

i s known the shape

of the self - induced step can be readily calculated.
T h e model circuits used in t his work will be discussed i n
T he first circui t con sists of a bridge of resi s tance

more detail.

R shunting one end of a l oss l ess microstrip of characteristi c
impedance Z

< R which is open at the other end.

The reactance

of the microstrip resonator at the bridge terminals is

= -z o cot (n - (1 )

X(w )
where

is the lowest resonant f r equency.

mode frequency n O ,

(3.25 )

If w

i s near the n-th

the equation (3.25) can be approximated as

X (w ) >:3

(3.26 )

where

o _ n

T he shape of the I-V characteristic in the ;neighborhood of
resonance is gotten by using the form (3.26 ) in equation (3.24),

-108V
qRI

where Q == nR/2Z

(3.27)

2 a

and On = nO

is the n-th resonant frequency.

The equation is a cubic equation for V (the time average voltagE\)
in tern1S of l/a (the din1ension1ess bias current) so that in
general n10re than one time average voltage may correspond to
a given bias current in the vicinity of resonance and the characteristic contains a negative resistance region.

Single valued I-V

characteristics V(a) or V(I) result if
aQ

110
< 1.5 2eqRI

(3.28)

The voltage deviation I:!, V due to the n-th resonance can be

identified £rOn1 equation (3 . 27) as
I:!,

qRI

a •

(3.29)

where

and,

as expected,

and

I:!,

qRI

for
I:!,

the width of th e step in the dOn1ain of the din1ensionless current
variable l/a is proportional to the paran1eter Q.
The second n10del circuit has a bridge of resistance R
connected in series at the center of a lossless n1icrostrip
segment of characteristic in1pedance Z

> R open at both ends .

-109The reactance of the microstrip segment at the bridge terminals
is

11 Wo

X(w )

where,

as before,

-ZZ cot ( - - )
Z (1

(3.30)

0. is the lowest resonant frequency.

Following

the procedure employed in the previous case a single valued I-V
characteristic results if
aQ<15
i1rJ
ZeqRI
where Q ==

l1Z /ZR.

(3.31)

The voltage deviation of the I-V charac-

te ristic due to the n-th re sonance is given by
t:J V

qRI

== + "2 a --------:[=-t:J-:--"'V~-q-:R:-:I:--------~J~Z

1 +4(Zn+l)2 Q 2

+ -v_c_(~ - ~ a) - 1

(3.32)

where only the lowest order term (in the dimensionless current
l/a) contributing to the self-induced step is shown.
the deviation t:J V

3.5.4

The sign of

at resonance is the reverse of (3.29).

Noise

In the above discussion the effects of noise on the shape
and magnitude of self-induced steps have been neglected.
as shown by Kirschman (Ref.

15),

However,

the bandwidth of Josephson

oscillation in proximity effect bridges is determined by the
amplitude of noise voltage across the bridge.

When environmental

sources of noise voltage are kept to a minimum the bandwidth of
the bridge oscillation is due to Johnson noise in the normal
current through the bridge.

According to Kirschman the oscil-

lation bandwidth for a proximity effect bridge is given by the
equation

-110-

where k ,

T, I ,

of the bridge,
quantum,

ili-0

(3.33)

are the Boltzmann constant, the temperature

the critical current of the bridge and the flux

respectively.

If the oscillation bandwidth M is much

smaller than the resonator bandwidth

fo/Q,

the noise will not

modify the self-induced step magnitude and shape appreciably,
provided that the theoretical I-V curve is single valued.
however,

If,

the bridge oscillation bandwidth exceeds the bandwidth

of the resonator,

part of the power spectrum of the oscillation

will not couple to the re sonator and the size !:; V
self-induced step will be reduced.

max

of the

At the same time the sharp

features of the self-induced steps will be washed out or broadened.
If the theoretical I-V curve is 'm ultivalued in the absence of

noise,

the effect of noise may be to induce transitions between

the several points on the I-V characteristic at a given bias
current 1.

Experimentally,

a single averaged voltage may be

measured in such a situation.

This would also reduce the step

slze from the maxi'mum value predicted from the noiseless model.
3.6

Results and Discussion
Both the size and the shape of self-induced steps observed

in the I-V characteristics of bridge-resonator systems lend
themselves to comparison with the theory presented in section

3.5.

To avoid the complications due to the possible complex

effects of noise (equation 3.33) and negative resistance (equations
3.28,

3. 31) on the shape of the steps only the low loaded Q

-111resonators were eITlployed for the con"lparison.

Two type 1

circuits and one type 2 circuit were thus chosen to obtain the
step size and shape data.
3.6.1

Step size

The type 1 circuits yielded multiple steps (at sequential
ITlode frequencies) enabling ITleasureITlents of step size over the
range 0.7 - 2.7 GHz.

Additional data on frequency dependence

were gotten by ITlodification of one circuit of each type extending
the data to 3.2 GHz.

The range of resonant frequencies and

bridge critical currents over which analyzable data could be
collected was liITlited by the sITlall step size (relative to noise) at
low critical currents Ic and high voltages V (V»

RIc),

and by

the extension of the lowest step into the critical current region
(r. Oj2e ~ RI ) at high critical currents I
or low first resonant

frequency O.

Between these liITlits the step size (defined as the

ITlaxiITluITl voltage deviation b. V

ITlax

froITl the I-V curve interpolated

froITl the nonresonant portions of the I-V characteristic) was
ITleasured by planiITletry frOITl the dV jdI vs. I characteristic.
The step size b.V
b.V

ITlax

thus ITleasured was norITlalized to

/RI and plotted against the normalized inverse bias current
max ' c
Figure s 12 -14 show the norITlalized data in cOITlparison

with the theoretical values (equations 3.15 - 3.20) based on the
b.vo alternative phase-supercurrent relations.
It is readily apparent that the data points fall between the

values predicted froITl the two alternative theories.

The data

points at the lower voltages (frequencies) and lower critical

-112currents are generally fitted better by the q = 1 curve (IS = rcsina)
whereas the data at th e highest frequencies and the highest
critical currents d eviate significantly towards the q = 1/2 curve
Ic
(I = 2-(1 + cos 6 )).
This trend is int erp reted as resulting from a true change
in the amplitude of Josephson oscillation in the proximity effect
bridges used in these experiments.

At low critical currents

(::::l0JJ.A) and low voltages (;S 3 JJ.V) the amplitude of the Josephson
oscillation is probably equal to the DC critical current of the
bridge .

At higher voltages and/or higher critical currents the

amplitude of supercurrent oscillation relative to the critical
current I

is progressive ly reduced.

These are indications both

from the present study and from the work of Franson (Ref. 14)
that at still highe r voltages (> 6 JJ. V) the reduction of the amplitude
of Josephson oscillation with voltage continues.

Franson deduced

the amplitude of the oscillating super current from microwave
imp e dance measurements in a Ta/W proximity effect bridge of
dimens ions similar to the bridges used in this work . except for
the length ( .R, (Franson)
frequency of 10 GHz,

= 0.5JJ.In,

= 0.8 - 1 JJ.m). At the
critical current I
= 40 JJ.A, bias current
.R, (Ganz)

= 150 JJ.A and resistance R = 0.17.n. he found that the amplitude

of th e Josephson oscillation was (0.62
frequencies (2eV /fi »

± 0.05) Ic.

At still higher

10 GHz) the behavior of proximity effect

bridges is a strong function of their geometry (Ref.

16).

In this

r egio n the relax ation time associated with the length of the bridge,
and heating due to dissipation become the important parameters

-113(ReI.

16).
The possible dependence of the phase-supercurrent relation

on the super current density in the bridge was predicted by Notarys
et al.

(Ref. 17) on the basis of a modified phase-slip model.

The

dependence of the phase -super current relation on the voltage V
across the bridge can be made plausible by the following argument.
Evidence was presented in Chapter 2 that the phase-supercurrent
relation at V = 0 is IS = Ic sino .

The phase-slip model predicts

that at finite voltages IS = 2c (1 + cos 0) holds.

It is likely that a

transition region exists at intermediate voltages V where the
amplitude of the supercurrent oscillation is intermediate between

and I

/2.

3.6.2

Step shape

An important and sensitive check on the validity of the
theoretical description of the origin of self-induced steps (section
3.5) is the comparison of the shape of experimentally observed
self-induced steps with the theoretical shape (equations 3.26 and
3.31),

The sensitivity of the test is increased by using the first

derivative d'1 /dI rather than the voltage V as the basis for
comparison.

The theoretical points were obtained by the

nUITlerical solution of the equation for the deviation 6. V(I)

6. '1(1)
=+,!.
RI
- 2

1 + 4Q

(3. 34)

RI - 6. . V

R(I '=F I 2/ 21 )

where the upper signs were used for type 2 circuits while the
lower signs were used for type 1 circuits.

The equation (3.34)

-114is an equivalent form of equations (3.29) and (3.32) for q = 1
The paraIneter I
center of the step and

is the loaded

is the bias current at the

of the n-th · step.

The

first derivative was also obtained numerically using the approximate form
d(lI V)

For type 2 circuits,

(3.35 )

where the characteristic impedance

Zo of the resonator can b e accurately determined from the known
dimensions only the current I
gotten by direct fitting.

at the center of the step was

The paraIneters I

fro ·m the portions of the dV /dI vs.
regions while 01 =

and R were Ineasured

I curve outside the resonant

f Zo/R .

For type 1 circuits where the characteristic iInpedance Z
was known only approximately,

the parameter 01 was obtained

by fitting as was the current In'

while On was calculated froIn

the equation
On

-- nO 1

(3.36)

Figure 15 shows the typical data fo r relatively low critical currents
and voltages.
currents (I

At higher voltages (V > 5 f.L V) and higher critical

> 10 f.LA) the agreement between the data and the

theory deteriorates,

presu·m ably due to the progressive decreas e

of the amplitude of the oscillating supercurrent compared to the
critical current I •

3. 7

Conclusion
dV
Self-induced steps have been observed in the I-V and dI

-115vs.

I characteristics of proxiInity effect bridges strongly coupled

to superconducting microstrip resonators.

The characteristic

impedanc e of the various resonators ranged fro ·m
with bridge resistances 0.1-0.2.!'\.,

10m.!'\. to 6 A

Steps corresponding to resonant

modes from 0.7 GHz to 10 GHz have been seen,

Small steps

generated by the second harmonic of Josephson oscillation have
also been observed in several samples.
For low critical currents (I

< lOf.LA) and low voltages

(V < 3 f.LV) the size and shape of self-induced steps agree with a
simple two fluid model as surning the phase - supercur rent relation
The deviation at higher voltage s and/or critical
currents towards the model which assumes the alternative phasesupercurrent relation IS =

-=(1 + cos 0) is interpreted to indicate

a pro gressive reduction (relative to the critical current I ) of
the amplitude of Josephson oscillation with increasing voltage V
and critical current I ,

-116-

Figure 12.

The norITlalized step size as a function of inverse
norITlalized bias current for a type 1 bridgeresonator . circuit (AF-l).
The theoretical curves
were calculated frOITl equation (3 . 16).
The frequencies f . f2 and f3 are the frequencies of the first
three step resonances.
ExperiITlental points are
labeled by the critical current I in p.A.
The bridge
resistance varied frOITl R
140 3.2 p.A
to R = 130 ITl.n. at I = 8.2 f-LA.
The ch~racteristic
iITlpedance of theITl!crostrip is estiITlated to be
Zo = (30±20) ITl.n..

et:

1-1

0 .1

0.2

0.1

0.2

Iel l

0.3

Fi gure 12

0.3I-AF-1
Ql fl = 0.73 GHz (1.5,uV)
+ f2= 1.4 GHz (2 .9,uV)
x f3= 2.1 GHz (4.3,uV)

0.4

· 0.6

--.l

......
......

-118-

Figure 13.

The norm.alized step size as a function of inverse
norm.alized bias current for a type 1 bridgeresonator circuit (AF-2).
The theoretical curves
were calculated fro"m. equation (3.16).
The second
set f , f7. resulted from. m.odification of the m.icrol
strip affer the first set of data was obtained.
The
bridge resistance varied from R == 135 m.n. at I ==
4.5 f.LA to R == 125 m.n. at I = 12.2 f.LA.
The cHaracteristic impedance Z
== (31l±20)mJ't as in Fig. 12.

+ f2= 1.8 GHz (3.7/-LV)

AF-2
(i) fl = 0.9 GHz ( 1.8 /-LV)
o fl = 1.3 GHz (2.7/-LV)

~""

0.1

0.2

J..

Figure 13

Iel l

0.3

6.8

0.4

~ .

?+_---------+12

4.5¥/'6.8 ,1< 12 1
~ 6 .8 __ --/ J.. •
v_-x"" ,.. ....0 12 2
...... ,.,...."
:.2}.6
.....
""./
-8
/0
.4
4.5°"'''' 6.8

0.1

@/~

4 .5~// ,,/"
8.4

0---4.6 .L--- /,,/

0.2r- x f2=2.6 GHz (5 .3/-LV)
I 0 f3=2.7 GHz (5.5/-LV)
et::

0.3

0.5

_---612.1

. ..

0.6

---r

12.21

--0

....

I-'

-120-

Figure 14.

The normalized step size as a function of inverse
normalized bias current for a type 2 bridgeresonator circuit (CIT-16BC).
The theoretical
curves were calculated from equation (3.20).
The
higher frequency data were obtained aftermicrostrip
modification.
The bridge resistance varied from
R = 100 rnA .at I = 3. 5 f!A to R = 90 rnA at I =
14. 5 f!A.
The cliaracteristic impedance coulcf be
determined accurately in this case to be Z
(240±20)mA.

ct:

t-i

0.1

0.1
Figure 14

o fl = 2.1 GHz (4.3 !-LV)
+ f, = 3.2 GHz (6.5j.LV)

CIT-16BC

Ic/ I

0.2

0.4

.....

I-'

-122-

Figure 15.

The comparison of theory (equations 3.34-3.36) with
experimental dV jell vs. I traces.
In graph A the
theory (dots) corresponds to Q = 4 as determined
from best fit (expected Q
n~j2Z
12±8).
The
in graph B was calculated a prio:= nZ j2R := 4±O.4.
The theory (dots) corresponds
to Q = 4.

-123-

Tn]
AF-l

0.3
0.2
0.1

elV

dB
[n]
CIT "16 Be

0.12

Figur e 15

1[1-1 A]

-124References
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P. L. Richards, F. Auracher, and T. Van Duzer, Proc.
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